If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms,
Question:
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
(a) 1/3
(b) 2/3
(c) 1/4
(d) 3/4
Solution:
(d) $3 / 4$
Let the terms of the G.P. be $a, a_{2}, a_{3}, a_{4}, a_{5}, \ldots, \infty$.
And, let the common ratio be $r$.
Now, $a+a_{2}=1$
$\therefore a+a r=1 \quad \ldots \ldots$ (i)
Also, $a=2\left(a_{2}+a_{3}+a_{4}+a_{5}+\ldots \infty\right)$
$\Rightarrow a=2\left(a r+a r^{2}+a r^{3}+a r^{4}+\ldots \infty\right)$
$\Rightarrow a=2\left(\frac{a r}{1-r}\right)$
$\Rightarrow 1-r=2 r$
$\Rightarrow 3 r=1$
$\Rightarrow r=\frac{1}{3}$
Putting the value of $\mathrm{r}$ in $(\mathrm{i})$ :
$a+\frac{a}{3}=1$
$\Rightarrow \frac{4 a}{3}=1$
$\Rightarrow 4 a=3$
$\Rightarrow a=\frac{3}{4}$